Command generation combining input shaping and smooth baseline functions

ABSTRACT

One preferred embodiment of the invention provides systems and methods for controlling a physical system by generating an input to the physical system that does not excite unwanted dynamics. Briefly described, one embodiment of the system among others, can be broadly summarized by as follows. A control entity generates a desired motion command for a physical system. A command generator then produces a shaped-smooth reference command for the physical system from the desired motion command that will cause the physical system to move in the desired motion without unwanted dynamics. Methods and other systems are also provided.

CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims priority to copending U.S. provisionalapplication entitled, “Command Generation Technique CombiningInput-Shaping and Smooth Functions for Residual Vibrations Reduction inComputer-Controlled Machines,” having serial No. 60/364,159, filed Mar.13, 2002, which is entirely incorporated herein by reference.

TECHNICAL FIELD

[0002] The present invention is generally related to mechanical systemsand, more particularly, is related to a system and method for reducingvibrations in mechanical systems.

BACKGROUND OF THE INVENTION

[0003] Unwanted vibration is a major problem that affects theperformance of many flexible mechanical systems. For example, when aflexible mechanical system is moved it has a tendency to vibrate. Thesevibrations can cause problems for the operator of the system. Thisvibration can cause damage to the system or surroundings or lowerproductivity by forcing the system to be moved slowly. Therefore it isadvantageous to reduce the level of vibration caused when thesestructures are moved. Such mechanical systems include coordinatemeasuring machines, wafer steppers, wafer handling robots, drillingmachines, disk head testers, hard disk drives, and robotic arms utilizedin space. For example, robotic arms, construction cranes, and satellitepositioning systems are often limited in their speed and accuracy byvibration.

[0004] In control systems, the commands used to perform a desired motioncan have a variety of shapes, and the shapes of these commands cangreatly affect system performance. In the field of command generationfor reducing mechanical vibrations, two fundamentally differenttechniques have often been opposed for achieving fast motions withminimum vibration: command smoothing and input shaping.

[0005] Command smoothing is a type of command generation that consistsof creating “smooth” profiles to move systems with compliance. Theintuitive concept behind these commands is that a flexible system shouldbe progressively accelerated to a maximum speed and then graduallydecelerated when approaching the desired setpoint so as to minimizemotion-induced vibrations. This technique counts on smooth transitionsbetween critical points of the trajectory to avoid exciting the flexiblemodes of the system. This smoothness is obtained via solving a set ofboundary conditions in velocity, acceleration, jerk, etc. Examples ofsmooth commands include S-curves, versines, and trigonometric functions.

[0006] Another solution for reducing vibrations is called commandshaping.

[0007] Command shaping attempts to negate any vibration induced by thereference command to the system by judiciously superimposing a delayedand scaled version of the command. Command shaping is not concerned withthe “smoothness” of the reference command. Instead, the choice of thedelayed and scaled command components depends on the known properties ofthe system such as natural frequency and damping ratio. Input shaping, aspecific subset of command shaping, is implemented by convolving asequence of impulses, an input shaper, with any desired motion commandto produce a reference command. By modifying the desired command in thisway, the input shaper acts to cancel the vibration induced by thedesired command.

[0008] Distinction has been made between command shaping and smoothcommand profiles on the basis of their shape and the system's response.In most cases, the smooth profiles have the effect of a low-pass filterwhile command shaping could be considered as notch filteringsuperimposed on whatever effect the reference command produces. Unlikecommand shaping, smooth commands usually fail to fully exploit the knownproperties of the system such as natural frequency and damping ratio.

[0009] These techniques generally work well on reducing residualvibrations in mechanical systems that predominately vibrate at one ortwo particular modes or frequencies. However, another important class ofvibratory systems has one or two dominant low modes and a range of highfrequencies. While S-curves, for example, suppress high frequencyvibrations due to their low pass filter qualities, the rise timeduration of the S-curve is a drawback, since it typically is severaltimes longer than that of a corresponding shaped command. Input shapingcan be used for high-mode limiting (HML) but requires extensivecomputation and is not very robust for unmodeled high modes.

[0010] Therefore, a robust and timely solution is desired for reducingvibrations for the class of vibratory systems featuring a wide range ofunmodeled high modes.

[0011] Ideally, the optimal solution would be to develop fast-risinglow-pass filtering commands that could both suppress low modes andensure unmodeled high modes do not degrade the system positioning. Thus,a heretofore unaddressed need exists in the industry to address theaforementioned deficiencies and inadequacies.

SUMMARY OF THE INVENTION

[0012] Preferred embodiments of the invention provide systems andmethods for controlling a physical system by generating an input to thephysical system that does not excite unwanted dynamics. Brieflydescribed, in architecture, one embodiment of the system, among others,can be implemented as follows. A control entity generates a desiredmotion command for a physical system. A command generator then producesa shaped-smooth reference command for the physical system from thedesired motion command that will cause the physical system to move inthe desired motion without unwanted dynamics.

[0013] The present invention can also be viewed as providing methods forcontrolling a physical system without exciting unwanted dynamics. Inthis regard, one embodiment of such a method, among others, can bebroadly summarized by the following steps: receiving a motion commandfor the physical system; and generating from the motion command ashaped-smooth reference command for the physical system that causes thephysical system to move according to the motion command while minimizingunwanted dynamics in the physical system.

[0014] Other systems, methods, features, and advantages of the presentinvention will be or become apparent to one with skill in the art uponexamination of the following drawings and detailed description. It isintended that all such additional systems, methods, features, andadvantages be included within this description and be within the scopeof the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Many aspects of the invention can be better understood withreference to the following drawings. The components in the drawings arenot necessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the present invention. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

[0016]FIG. 1A is a block diagram of a control system of one embodimentof the present invention.

[0017]FIG. 11B is a block diagram of one embodiment of the commandgenerator system shown in FIG. 1A.

[0018]FIG. 2A is a graph of an example frequency spectrum for a systemtargeted by the command generator system of FIG. 1B.

[0019]FIG. 2B is a graph of an example frequency spectrum for a systemtargeted by the command generator system of FIG. 1B.

[0020]FIG. 3 is a graph of an example frequency span of the residualvibrations that are targeted by the command generator system of FIG. 1B.

[0021]FIG. 4 is a graph of the command rise times produced by thecommand generator system of FIG. 1B for commands shaped with positiveinput shapers.

[0022]FIG. 5 is a graph of rise time penalties generated by the commandgenerator of FIG. 11B for commands shaped with positive input shapers.

[0023]FIG. 6 is a graph of the command rise times produced by thecommand generator of FIG. 1B for commands shaped with negative inputshapers.

[0024]FIG. 7 is a graph of rise time penalties generated by the commandgenerator of FIG. 11B for commands shaped with negative input shapers.

[0025]FIG. 8 is a graph of the command rise times produced by thecommand generator of FIG. 1B for negative shaped-smooth commands versusa positive shaped step command.

[0026]FIG. 9 is a graph of rise time penalties generated by the commandgenerator of FIG. 1B for negative shaped-smooth commands in relation toa positive shaped step command.

[0027]FIG. 10 is a flow chart describing the functionality of apreferred implementation of the control system of FIG. 1A.

[0028]FIG. 11 is a flow chart describing the functionality of apreferred implementation of the command generator system of FIG. 1B forminimizing unwanted dynamics occurring at a low mode and a range of highmodes.

[0029]FIG. 12 is a flow chart describing one embodiment of a method forselecting a shaped-smooth command generated by the command generator ofFIG. 1B.

[0030]FIG. 13 is a graph of overshoot vs. system frequency for varioussmooth commands that may be utilized by the command generator of FIG.1B.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0031] The performance of mechanical systems depends on numerousvariables such as the mechanical design, the operating environment, andthe control system. The most important influence on performance variesfrom system to system and may change over time, or with the task beingperformed. However, the control system is almost always an importantfactor in system performance. Given the increasing use of computers tocontrol mechanical systems and the trend toward faster, lighter, andmore flexible structures, control system design and implementation willcontinue to gain in importance. The control system must performfunctions such as positioning, trajectory tracking, suppression ofresidual vibration, obstacle avoidance, and disturbance rejection.

[0032]FIG. 1A shows a block diagram of a control system of oneembodiment of the present invention. The control system 100 includes aphysical plant 10, feedback control 20, feedforward block 30, controlentity 40, and a command generator system 50. Other embodiments of theinvention may not necessary include feedback 20 and feedforward elements30.

[0033] For the embodiment represented in FIG. 1A, unwanted vibration inthe mechanical system of the physical plant 10 may be treated by variousmethods. The physical plant 10 can be modified to make it less flexible,or the feedback control 20 can be tuned to damp out vibration. Thefeedforward block 30 can be used to inject control effort into the loop,so as to negate vibration. The fourth option is the command generatorsystem 50.

[0034] The desired motion command D(t) of the system 10 is fed into thecommand generator system 50 that transforms the desired motion commandD(t) into a reference command r(t). If the reference commands have anappropriate shape, then they will produce the desired motion, whilereducing the detrimental effects of flexibility.

[0035] For one preferred embodiment of the invention, as shown in FIG.11B, the command generator system 50 generates a fast-rising inputcommand that suppresses low modes while ensuring unmodeled high modes donot degrade the system performance. This is achieved by convolving afast-rising smooth command profile as a baseline function 210 with aninput shaper 220 to take advantages of the respective properties ofeach. Stated differently, input shaping is used for vibrationsuppression at the low frequencies and command smoothing for noisesuppression at high frequencies. The roll-off frequency of the commandsmoothing is set near the onset of the high frequency dynamics.Therefore, the command smoothing does not cause a large time lag in thesystem. As shown in FIG. 1B, the desired motion command D(t) isconverted to a baseline reference command b(t) via the smooth commandconverter 210 and then filtered by the input shaper 220 to produce thereference command r(t).

[0036] Shown in FIG. 2A are possible frequency spans that each componentof the command generator system 50 suppresses for a preferredembodiment. As shown in FIG. 2A, for a targeted physical system 10,there is residual vibration occurring at a low frequency or mode, or acouple of low frequencies. Then, there is a gap with no significantvibration modes. This gap is followed by high frequencies or modes thatdo experience residual vibrations. The narrow frequency range around COLis dealt with by the input shaper (e.g., zero vibration (ZV) shaper)while any high modes starting at ω_(H) are attentuated by the smoothprofile (e.g., S-curve). FIG. 2B shows a similar case where there aretwo low frequencies. Accordingly, the command generator system 50suppresses vibration where there are a few low frequencies and a groupof high frequencies. For example, disk drives are of this type ofsystem, along with satellites and many types of manufacturing machinesthat have fairly complicated dynamics.

[0037] The main attribute of most smooth profiles is their low-passfiltering characteristic which can minimize residual vibrations at highfrequencies. Common smooth command profiles include S-curves, versines,trigonometric transition functions, and cam polynomials. These smoothprofiles rely on their smoothness to minimize the excitation of theflexible modes. Generally, only the command rise time of the smoothcommands can be adjusted to significantly affect the frequencysuppression. These methods by themselves usually fail to fully exploitthe known properties of the system such as natural frequency and dampingratio and instead simply provide a low pass filtering effect.

[0038] With a command generator 50 that produces fast-rising low-passfiltering reference commands r(t), low modes in physical system 10 willbe suppressed while simultaneously unmodeled high modes are ensured tonot degrade the system positioning. This overcomes the typical drawbackto S-curves and other smooth commands regarding their slow rise times.By combining a fast rising S-curve with an input shaper, a referencecommand r(t) is produced that minimizes low and high frequencyvibrations in a short amount of time. For example, the rise time penaltyof a ZV-shaped-S-curve command produced by one embodiment of theinvention compared to a conventional ZV input shaper is really small andyet, the performance of the system 10 is significantly better, since aZV-shaper does not attenuate high frequencies when present.

[0039] Note, mechanical systems 10, especially flexible systems, mayhave a large, possibly infinite number of modes. For modeling reasons,this number is often reduced to a few dominant low modes and some rangeof higher modes. Because the time required to cancel vibrations is verydependent on the lower modes of a system, it is useful to relate thecommand rise time to the mode ratios before selecting a referencecommand r(t).

[0040] Accordingly, FIG. 3 shows the frequency span for a system with asingle low mode, f_(low), and high modes ranging from f_(high) toαf_(high) α>1, where a specifies the span of high frequencies that acontrol system must suppress. However, the most important parameter forchoosing the reference command r(t) is the ratio of the lowest high modefrequency divided by the low mode frequency (f_(high)/f_(low)). As thismode ratio increases, the rise time gap between shaped-smooth commandsand input-shaped commands diminishes. This effect can be observed inFIG. 4 for a sample system having a 1-Hz low mode and high modevibrations in the frequency range from r, the mode ratio, to αr, where ais 3 in this case. The figure compares the command rise times of theZV-shaped S-curve, the ZV-shaped versine, and the ZV-HML shaped step. Inthe instance of a mode ratio of 10, the ZV-shaped S-curve and theZV-shaped versine are only 10 and 15% longer than the ZV-HML-shaped stepinput.

[0041]FIG. 5 offers another valuable rise time comparison as it showsthe rise time penalty of the ZV-shaped smooth commands that may begenerated by the command generator 50 over step inputs convolved with aZV-HML shaper for the sample system. As mentioned above, for a moderatio of 10, the ZV-shaped S-curve and the ZV-shaped versine are only 10and 15% longer than the ZV-HML-shaped step input. For a mode ratio of 2,however, the penalty is more than doubled.

[0042] The same trends can be observed with negative input shapers, asdemonstrated in FIGS. 6 and 7. But, in this case, utilizing input-shapedsmooth commands (UMZV-Shaped S-Curve and UMZV-Shaped Versine) overHML-shaped step commands (UMZV-Shaped step input) is slightly morecostly relative to the positive impulse case.

[0043] The rise time drawback of shaped-smooth commands may be of littlecost in regard to some advantages from using shaped-smooth commands. Onepotential benefit is that no optimization is needed to shape smoothprofiles with single-mode or simple two-mode shapers. Furthermore, theduration of UMZV and ZV-HML step inputs may have to be lengthened due tohardware limitations. Indeed it sometimes happens that the hardwaresampling rate is not high enough to accurately convolve any command witha ZV-HML shaper. The remedy is then to generate an optimization tolocate the impulses at multiple integers of the sampling period, hencepossibly increasing the shaper duration slightly.

[0044] Due to their low-pass filtering properties, smooth commands donot excite potential unmodeled high modes beyond αf_(high), even whencombined with negative shapers. Thus by pairing smooth commands withnegative input shapers, for example, unmodeled high modes beyondαf_(high) are not excited. In other words, for systems with unmodeledhigh modes, the duration of ZV-HML shapers can be preferably comparedagainst the rise time UMZV-shaped smooth commands. FIG. 8 shows thatabove mode ratios of 3.5 and 5 respectively, the UMZV-shaped versine andS-curve (as utilized in some preferred embodiments) become shorter thanthe ZV-HML shaper. Even for low mode ratios, their time penalty is notoverly large as demonstrated in FIG. 9.

[0045] The overall operation 1000 of the control system 100 will be nowdescribed with reference to FIG. 10, which depicts the functionality ofa preferred implementation of the control system 100. First, in step1010, important characteristics of the mechanical system (physicalplant) 10 are identified, such as the modes of vibration for themechanical system, and damping ratios, for example. Then, as shown instep 1020, a steep (fast-rising) smooth baseline reference command b(t)is generated from the desired motion command D(t). The shape of thesmooth baseline reference command b(t) is designed to have a short risetime in order to only suppress high frequency vibrations. Next in step1030, based upon the characteristics of the mechanical system 10, thebaseline reference command b(t) is convolved with an input shaper(impulse sequence) to produce a reference command r(t) for themechanical system. The reference command r(t) is then delivered to themechanical system, as shown in step 1040.

[0046] As previously discussed, one particular type of system that maybe targeted by this process 1000 is one where there is a low frequency,or a couple of low frequencies, and some gap, and then a range of highfrequency vibrations (as shown in FIG. 2A). One implementation ofaddressing this problem is shown in FIG. 11. Here, it is identified thatthe mechanical system 10 features a low mode of vibration and range ofvibrations in a high frequency range, as shown in step 1110. The desiredmotion command D(t) for the system is then converted into a smoothcommand, such as a S-curve command, b(t) that has a short rise time andis designed to suppress high frequency vibrations above the lower end ofthe high mode range, as shown in step 1120. The S-curve command b(t) isthen convolved with a ZV shaper (that is designed to remove vibration atlow modes) to produce the reference command r(t) of the mechanicalsystem 10, as represented in step 1130. From the ZV shaper filtering,reference command r(t), once applied, eliminates the vibrations at thelow mode of vibration and from the S-Curve conversion, reference commandr(t) removes high frequency vibrations, as shown in step 1140.

[0047] FIGS. 5-9 demonstrate that the process 1000 of using smoothcommand shaping is preferable for a system having high modes that aresignificantly higher than the low mode. Yet, in some other instances,choosing between ZV-shaped S-Curves and ZV-HML-shaped step inputs, forexample, may be less obvious, since the choice depends on factors suchas the various modes of the system, the uncertainty on the high modes,etc. Accordingly, a method 1200 for selecting the appropriateshaped-smooth command for one embodiment of the invention is shown inFIG. 12.

[0048] The method 1200 accounts for the command rise time, thepossibility of unmodeled high modes, and the complexity of generatingthe-shaped-smooth command. As such, the following rules apply to themethod shown in FIG. 12:

[0049] 1) In the event of expected unmodeled high modes, only positiveinput shapers are used on step inputs, as negative input shapers mayexcite those modes.

[0050] 2) For smooth commands with equal rise times, the most efficientlow pass filter is used. This statement is motivated by the fact thatall smooth commands do not attenuate vibration by the same amount, pastthe rolloff frequency as observed in FIG. 13. For instance, with thesame rise time, trigonometric transition functions are more effectivelow pass filters than S-curves.

[0051] 3) As HML-shaped step inputs can be shorter than shaped-smoothcommand, the user must also decide whether HML input shapers are worththe optimization effort or not. Although this decision will vary fromuser to user, it is considered that a smooth command rise time penaltyof 20% is acceptable given the simplicity of generating smooth profiles.

[0052] Taking the above into account, the first step 1205 of FIG. 12 isidentifying the mode ratio (or gap) between the low frequency dynamicsand the high frequency dynamics of the mechanical system 10 of interest.Then, it is determined whether the mechanical system 10 has unmodeledhigh frequencies, as shown in step 1210. If there are none, then atraditional input shaping command, such as a UMZV-HML Shaper, is usedfor a mode ratio that is less than 5, as shown in steps 1220-1225. Ifthe mode ratio is greater than 5 and less than 10 then a UMZV-ShapedVersine command is used, as shown in steps 1230 and 1260. Else, if themode ratio is greater than 10, a UMZV-Shaped Trigonometric TransitionFunction is used, as shown in step 1270.

[0053] On the other hand, if it is determined that there are unmodeledhigh frequencies and the mode ratio is less than 2.5, the appropriateshaped-smooth reference command is a UMZV-Shaped Versine signal, asshown in steps 1210, 1250, and 1260. Otherwise, if there are unmodeledhigh frequencies and the mode ratio is greater than 2.5, then anappropriate shaped-smooth reference command for this implementation ofthe invention is a UMZV-Shaped Trigonometric Function, as shown in steps1250 and 1270.

[0054] Note, the utilization of UMZV input shapers with smooth commandshaving low pass filtering characteristics advantageously andbeneficially reduces residual vibrations in systems with higher ordermodes. This is particularly preferential when actuator limits preventthe use of step or fast-rising ramp inputs in certain mechanicalsystems.

[0055] In alternative embodiments of the invention, other decisionprocesses for systems with a low mode and a range of higher modes arepossible besides those shown in FIG. 12. For example, the decisionblocks 1220, 1230, 1250 based on mode ratio could contain different moderatio values. However, the method shown in FIG. 12 should prove to beapplicable to many of those systems.

[0056] The enhanced vibration suppressing capabilities of theabove-described embodiments of the present invention advantageouslydemonstrates the effectiveness of intelligently combining input shapingand command smoothing to reduce residual vibrations on systems with lowmodes and a range of higher modes. The notch filtering properties ofinput shaping suppress the low modes individually while keeping theduration of the command as short as possible. Diversely, smoothcommands, which are essentially low-pass filters, attenuate potentialhigh-mode excitations. For such systems, the association of the twotechniques is often a better choice over the selection of input shapersthat suppress the low modes and limit the high modes below a tolerablelevel. Further, the computational demands of the above-describedembodiments are relatively small as compared to other techniques.

[0057] The command generator 50 of a representative embodiment of thepresent invention can be implemented in hardware, software, firmware, ora combination thereof. In the. preferred embodiment(s), the commandgenerator 50 is implemented in hardware with any or a combination of thefollowing technologies, which are all well known in the art: a discretelogic circuit(s) having logic gates for implementing logic functionsupon data signals, an application specific integrated circuit (ASIC)having appropriate combinational logic gates, a programmable gatearray(s) (PGA), a field programmable gate array (FPGA), etc. Inalternative embodiment(s), the command generator 50 is implemented insoftware or firmware that is stored in a memory and that is executed bya suitable instruction execution system.

[0058] The flow charts of FIGS. 10-12 show the functionality andoperation of a possible implementation of the control system of thepresent invention. In this regard, each block represents a module,segment, or portion of code, which comprises one or more executableinstructions for implementing the specified logical function(s). Itshould also be noted that in some alternative implementations, thefunctions noted in the blocks may occur out of the order noted in thefigures. Any process descriptions or blocks in flow charts should beunderstood as representing modules, segments, or portions of code whichinclude one or more executable instructions for implementing specificlogical functions or steps in the process, and alternate implementationsare included within the scope of the preferred embodiment of the presentinvention in which functions may be executed out of order from thatshown or discussed, including substantially concurrently or in reverseorder, depending on the functionality involved, as would be understoodby those reasonably skilled in the art of the present invention.

[0059] It should be emphasized that the above-described embodiments ofthe present invention, particularly, any “preferred” embodiments, aremerely possible examples of implementations, merely set forth for aclear understanding of the principles of the invention. Many variationsand modifications may be made to the above-described embodiment(s) ofthe invention without departing substantially from the principles of theinvention. All such modifications and variations are intended to beincluded herein within the scope of this disclosure and the presentinvention and protected by the following claims.

Therefore, having thus described the invention, at least the followingis claimed:
 1. A system for controlling a physical system by generatingan input to the physical system that does not excite unwanted dynamicscomprising: a command generator responsive to a motion command togenerate a shaped-smooth command as input to the physical system forcontrolling the physical system while suppressing unwanted dynamics inthe physical system.
 2. The system of claim 1, further comprising: amoveable structure within the physical system; and a control devicefunctionally connected to the moveable structure and operable togenerate the motion command for directing a desired movement of themoveable structure.
 3. The system of claim 1, wherein the unwanteddynamics include residual vibrations occurring at at least one low modeand a range of high modes.
 4. The system of claim 1, the commandgenerator further comprising: an apparatus for converting the motioncommand to a smooth baseline reference command; an apparatus forconvolving the smooth baseline reference command with a particularimpulse sequence to produce the shaped-smooth command.
 5. The system ofclaim 4, wherein the smooth baseline reference command is designed tohave a short rise time to minimize unwanted dynamics at a range of highmodes and not at low modes.
 6. The system of claim 4, wherein theimpulse sequence is designed to remove unwanted dynamics at a particularlow mode based upon characteristics of the physical system.
 7. Thesystem of claim 4, wherein the impulse sequence is designed to removeunwanted dynamics at a plurality of particular low modes based uponcharacteristics of the physical system.
 8. The system of claim 4,wherein the smooth baseline reference command comprises a S-curveprofile.
 9. The system of claim 4, wherein the impulse sequence containsnegative impulses.
 10. The system of claim 1, wherein the commandgenerator generates a particular shaped-smooth command based upon aparticular mode ratio of unwanted dynamics.
 11. The system of claim 1,wherein the command generator suppresses the unwanted dynamics below apredefined level.
 12. A system for controlling a physical system bygenerating an input to the physical system that does not excite unwanteddynamics comprising: means for generating a motion command for thephysical system; and means for generating a shaped-smooth command asinput to the physical system from the motion command that suppressesunwanted dynamics in the physical system.
 13. The system of claim 12,the means for generating a shaped-smooth command comprising: means forconverting the motion command to a smooth command; and means forconvolving the smooth command with an impulse sequence to produce theshaped-smooth command.
 14. The system of claim 12, further comprising:means for identifying a mode ratio of unwanted dynamics in the physicalsystem, wherein the motion command is converted to a particularshaped-smooth command based upon the mode ratio.
 15. A method forcontrolling a physical system by generating an-input to the physicalsystem that does not excite unwanted dynamics comprising: receiving amotion command for the physical system; and generating a shaped-smoothcommand as input to the physical system from the motion command thatsuppresses unwanted dynamics in the physical system.
 16. The method ofclaim 15, the generating step comprising: identifying importantcharacteristics about the dynamics of the physical system; and designingthe shaped-smooth command based upon these important characteristics.17. The method of claim 16, wherein the important characteristicsinclude a low mode of residual vibration and a high mode range ofresidual vibration.
 18. The method of claim 16, wherein the importantcharacteristics include a plurality of low modes of residual vibrationand a high mode range of residual vibration.
 19. The method of claim 16,wherein the important characteristics include a damping ratio.
 20. Themethod of claim 16, wherein the important characteristics include a moderatio of unwanted dynamics that determines a particular type ofshapedsmooth command that is to be generated for a particular moderatio.
 21. The method of claim 15, the generating step comprising:converting the motion command to a smooth command; and convolving thesmooth command with an impulse sequence to produce the shaped-smoothcommand.
 22. The method of claim 21, wherein the smooth command isdesigned to have a short rise time to minimize unwanted dynamics at highmodes and not at low modes.
 23. The method of claim 21, wherein thesmooth command comprises a Scurve profile.
 24. The method of claim 21,wherein the impulse sequence contains negative impulses.
 25. The methodof claim 15, the generating step comprising: identifying a mode ratiofrom the dynamics of the physical system; and for a particular moderatio, producing a particular shaped-smooth command that minimizesunwanted dynamics characterized by the particular mode ratio.
 26. Themethod of claim 15, the generating step comprising: identifying a moderatio from the dynamics of the physical system; generating a shaped stepinput command if the mode ratio is less than a mode ratio parameter; andgenerating a shaped-smooth command if the mode ratio is more than themode ratio parameter.
 27. The method of claim 15, the generating stepcomprising: identifying a mode ratio from the dynamics of the physicalsystem; generating a shaped step input command if the mode ratio is lessthan a first mode ratio parameter; generating a first shaped-smoothcommand if the mode ratio is more than the first mode ratio parameterand less than a second mode ratio parameter; and generating a secondshaped-smooth command if the mode ratio is more than the second moderatio parameter.
 28. The method of claim 15, wherein the unwanteddynamics are suppressed below a predefined level.
 29. A computerreadable medium having a computer program for controlling a physicalsystem by generating an input to the physical system that does notexcite unwanted dynamics, the program for performing the steps of:receiving a motion command for the physical system; and generating ashaped-smooth command as input to the physical system from the motioncommand that suppresses unwanted dynamics in the physical system. 30.The medium of claim 29, the generating step comprising: identifyingimportant characteristics about the dynamics of the physical system; anddesigning the shaped-smooth command based upon these importantcharacteristics.
 31. The medium of claim 29, the generating stepcomprising: converting the motion command to a smooth command; andconvolving the smooth command with an impulse sequence to produce theshaped-smooth command.
 32. The medium of claim 29, the generating stepcomprising: identifying a mode ratio from the dynamics of the physicalsystem; and for a particular mode ratio, producing a particularshaped-smooth command that minimizes unwanted dynamics characterized bythe particular mode ratio.
 33. The medium of claim 29, the generatingstep comprising: identifying a mode ratio from the dynamics of thephysical system; generating a shaped step input command if the moderatio is less than a mode ratio parameter; and generating ashaped-smooth command if the mode ratio is more than the mode ratioparameter.
 34. The medium of claim 29, the generating step furthercomprising: identifying a mode ratio from the dynamics of the physicalsystem; generating a shaped step input command if the mode ratio is lessthan a first mode ratio parameter; generating a first shaped-smoothcommand if the mode ratio is more than the first mode ratio parameterand less than a second mode ratio parameter; and generating a secondshaped-smooth command if the mode ratio is more than the second moderatio parameter.